Advanced Risk Management I

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Advanced Risk Management I

• Lecture 3
• Market risk transfer Hedging

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Choice of funding

• Assume you want to fund an investment. Then, one
  first has to decide the funding. What would you
  recommand?
• What are the alternatives?
• Fixed rate funding
• Floating rate fundign
• Structured funding (with derivatives)

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Fixed rate funding

• Pros future cash flows are certain
• Cons future market value of debt certain
• Fixed rate funding risks
• In case of buy-back lower interest rates would
  imply higher cost
• If the investment cash flows are positively
  correlated with interest rates, when rates go
  down the value of the asset side decreases and
  the value of liabilities decreases.

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Floating rate funding

• Pros stable market value of debt
• Cons future cash flows are uncertain
• Floating rate funding risk
• An increase of the interest rates can induce a
  liquidity crisis
• If the investment cash flows are negatively
  correlated with interest rates, when rates go up
  the value of the asset side decreases and the
  value of liabilities increases.

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Intermediate funding choices

• Plain fixed and floating funding presents extreme
  risks of opposite kind swing of mark-to-market
  value vs swing of the future cash-flows.
• Are there intermediate choices?
• Issuing part of debt fixed and part of it
  floating
• Using derivatives automatic tools to switch from
  fixed to floating funding or vice versa.

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Why floating coupons stabilize the value of
debt?

• Intuitively, if coupons are fixed, the increase
  in interest rates reduces the present value of
  future cash flows
• Il coupons are designed to increase with interest
  rates, then the effect of an interest rate upward
  shock on the present value of future cash flows
  is mitigated by the increase in future coupons
• If coupons are designe to decrease with interest
  rates, then the effect of an interest rate upward
  shock on the present value of future cash flows
  is reinforced by the decrease in future coupons
  (reverse floater)

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Indexed (floating) coupons

• An indexed coupon is determined based on a
  reference index, typically an interest rates,
  observed at time ?, called the reset date.
• The typical case (known as natural time lag) is a
  coupon with
• reference period from ? to T
• reset date ? and payment date T
• reference interest rate for determination of the
  coupon
• i(? ,T) (T ? ) 1/v (? ,T) 1

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Replicating portfolio

• What is the replicating portfolio of an floating
  coupon, indexed to a linear compounded interest
  rate for one unit of nominal?
• Notice that at the reset date ? the value of the
  coupon, determined at time ? and paid at time T,
  will be given by
• v (? ,T) i(? ,T) (T ? ) 1 v (? ,T)
• The replicating portfolio is then given by
• A long position (investment) of one unit of
  nominal available at time ?
• A short position (financing) for one unit of
  nominal available at time T

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Cash flows of a floating coupon

• Notice that a floating coupon on a nominal amount
  C corresponds to a position of debt (leverage)

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No arbitrage priceindexed coupons

• The replicating portfolio enables to evaluate the
  coupon at time t as
• indexed coupons v(t,?) v(t,T)
• At time ? we know that the value of the position
  is
• 1 v(?,T) v(?,T) 1/ v(?,T) 1
• v(?,T) i(?,T)(T ?)
• discount
  factor X indexed coupon
• At time t the coupon value can be written
• v(t,?) v(t,T) v(t,T)v(t,?) / v(t,T) 1
• v(t,T) f(t,?,T)(T ?)
• discount
  factor X forward rate

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Indexed coupons some caveat

• It is wrong to state that expected future coupons
  are represented by forward rates, or that
  forward rates are unbiased forecasts of future
  forward rates
• The evaluation of expected coupons by forward
  rates is NOT linked to any future scenario of
  interest rates, but only to the current interest
  rate curve.
• The forward term structure changes with the spot
  term structure, and so both expected coupons and
  the discount factor change at the same time (in
  opposite directions)

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Indexed cash flows

• Let us consider the time schedule
• ?t,t1,t2,tm?
• where ti, i 1,2,,m 1 are coupon reset
  times, and each of them is paid at ti1.
• t is the valuation date.
• It is easy to verify that the value the series of
  flows corresponds to
• A long position (investment) for one unit of
  nominal at the reset date of the first coupon
  (t1)
• A short position (financing) for one unit of
  nominal at the payment date of the last coupon
  (tm)

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Floaters preserve the value of debt

• A floater is a bond characterized by a schedule
• ?t,t1,t2,tm?
• at t1 the current coupon c is paid (value
  cv(t,t1))
• ti, i 1,2,,m 1 are the reset dates of the
  floating coupons are paid at time ti1 (value
  v(t,t1) v(t,tm))
• principal is repaid in one sum tm.
• Value of coupons cv(t,t1) v(t,t1) v(t,tm)
• Value of principal v(t,tm)
• Value of the bond
• Value of bond Value of Coupons Value of
  Principal
• cv(t,t1) v(t,t1)
  v(t,tm) v(t,tm)
• (1 c) v(t,t1)
• A floater is financially equivalent to a short
  term note.

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EserciseReverse floater

• A reverse floater is characterized by a time
  schedule
• ?t,t1,t2,tj, tm?
• From a reset date tj coupons are determined on
  the formula
• rMax ? i(ti,ti1)
• where ? is a leverage parameter.
• Principal is repaid in a single sum at maturity

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Natural lag

• In this analysis we have assumed (natural lag)
• Coupon reset at the beginning of the coupon
  period
• Payment of the coupon at the end of the period
• Indexation rate is referred to a tenor of the
  same length as the coupon period (example,
  semiannual coupon indexed to six-month rate)
• A more general representation
• Expected coupon forward rate
• convexity adjustment timing adjustment
• It may be proved that only in the natural lag
  case
• convexity adjustment timing adjustment 0

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Managing interest rate risk

• Interest rate derivatives FRA and swaps.
• If one wants to change the cash flow structure,
  one alternative is to sell the asset (or buy-back
  debt) and buy (issue) the desired one.
• Another alternative is to enter a derivative
  contract in which the unwanted payoff is
  exchanged for the desired one.

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Forward rate agreement (FRA)

• A FRA is the exchange, decided in t, between a
  floating coupon and a fixed rate coupon k, for an
  investment period from ? to T.
• Assuming that coupons are determined at time ?,
  and set equal to interest rate i(?,T), and paid,
  at time T,
• FRA(t) v(t,?) v(t,T) v(t,T)k
• v(t,T) v(t,?)/ v(t,T) 1 k
• v(t,T) f(t,?,T) k
• At origination we have FRA(0) 0, giving k
  f(t,?,T)
• Notice that market practice is that payment
  occurs at time ? (in arrears) instead of T (in
  advance)

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Swap contracts

• The standard tool for transferring risk is the
  swap contract two parties exchange cash flows in
  a contract
• Each one of the two flows is called leg
• Examples of swap
• Fixed-floating plus spread (plain vanilla swap)
• Cash-flows in different currencies (currency
  swap)
• Floating cash flows indexed to yields of
  different countries (quanto swap)
• Asset swap, total return swap, credit default
  swap

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Swap parameters to be determined

• The value of a swap contract can be expressed as
• Net-present-value (NPV) the difference between
  the present value of flows
• Fixed rate coupon (swap rate) the value of fixed
  rate payment such that the fixed leg be equal to
  the floating leg
• Spread the value of a periodic fixed payment
  that added to to a flow of floating payments
  equals the fixed leg of the contract.

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Plain vanilla swap (fixed-floating)

• In a fixed-floating swap
• the long party pays a flow of fixed sums equal
  to a percentage c, defined on a year basis
• the short party pays a flow of floating payments
  indexed to a market rate
• Value of fixed leg
• Value of floating leg

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Swap rate

• In a fixed-floating swap at origin
• Value fixed leg Value floating leg

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Swap rate

• Representing a floating cash flow in terms of
  forward rates, a swap rate can be seen as a
  weghted average of forward rates

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Swap rate

• If we assume ot add the repayment of principal to
  both legs we have that swap rate is the so called
  par yield (i.e. the coupon rate of a fixed coupon
  bond trading at par)

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Bootstrapping procedure
Assume that at time t the market is structured on
m periods with maturities tk t k, k1....m,
and assume to observe swap rates on such
maturities. The bootstrapping procedure enables
to recover discount factors of each maturity
from the previous ones.
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Forward swap rate

• In a forward start swap the exchange of flows
  determined at t begins at tj.
• Value fixed leg Value floating leg

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Swap rate summary

• The swap rate can be defined as
• A fixed rate payment, on a running basis,
  financially equivalent to a flow of indexed
  payments
• A weighted average of forward rates with weights
  given by the discount factors
• The internal rate of return, or the coupon, of a
  fixed rate bond quoting at par (par yield curve)

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Forward contracts

• The long party in a forward contract defines at
  time t the price F at which a unit of the
  security S will be purchased for delivery at time
  T
• At time T the value of the contract for the long
  party will be S(T) - F

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Derivatives and leverage

• Derivative contracts imply leverage
• Alternative 1
• Forward 10 000 ENEL at 7,2938 , 2 months
• 2 m. later Value 10000 ENEL 72938
• Alternative 2
• Long 10 000 ENEL spot with debt 72938 for
  repayment in 2 months.
• 2 m. later Value 10000 ENEL 72938

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Hedging and speculation

• Assume you have 10000 ENEL stocks in your
  portfolio, and say that
• You go short a forward contract on 10 000 ENEL
  for 72 938. Then, using the replicating
  portfolio, we have that the 10 000 ENEL are
  virtually removed from the portfolio, and the
  portfolio is worth the present discounted value
  of 72 938.
• You go long a forward contract on 10 000 ENEL for
  72 938. Then, using the replicating portfolio you
  are exposed to 20 000 ENEL with a debt of 72 398
  euros. This is the leverage effect.

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Non linear contracts options

• Call (put) European gives at time t the right,
  but not the obligation, to buy (sell) at time T
  (exercise time) a unit of S at price K (strike or
  exercise price).
• Payoff of a call at T max(S(T) - K, 0)
• Payoff of a put at T max(K - S(T), 0)

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Black Scholes model

• Black Scholes model is based on the assumption
  of normal distribution of returns. The model is
  in continuous time. Recalling the forward price
  F(Y,t) Y(t)/v(t,T)

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Put-Call Parity

• Portfolio A call option v(t,T)Strike
• Portfolio B put option underlying
• Call exercize date T
• Strike call Strike put
• At time T
• Value A Value B max(underlying,strike)
• and no arbitrage implies that portfolios A and B
  must be the same at all t lt T, implying
• Call v(t,T) Strike Put Undelrying

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Put options

• Using the put-call parity we get
• Put Call Y(t) v(t,T)K
• and from the replicating portfolio of the call
• Put (? 1)Y(t) v(t,T)(K W)
• The result is that the delta of a put option
  varies between zero and 1 and the position in
  the risk free asset varies between zero and K.

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Structuring principles

• Questions
• Which contracts are embedded in the financial or
  insurance products?
• If the contract is an option, who has the option?

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Who has the option?

• Assume the option is with the investor, or the
  party that receives payment.
• Then, the payoff is
• Max(Y(T), K)
• that can be decomposed as
• Y(T) Max(K Y(T), 0) or
• K Max(Y(T) K, 0)

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Who has the option?

• Assume the option is with the issuer, or the
  party that makes the payment.
• Then, the payoff is
• Min(Y(T), K)
• that can be decomposed as
• K Max(K Y(T), 0) or
• Y(T) Max(Y(T) K, 0)

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Convertible

• Assume the investor can choose to receive the
  principal in terms of cash or n stocks of asset S
• max(100, nS(T))
• 100 n max(S(T) 100/n, 0)
• The contract includes n call options on the
  underlying asset with strike 100/n.

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Reverse convertible

• Assume the issuer can choose to receive the
  principal in terms of cash or n stocks of asset S
• min(100, nS(T))
• 100 n max(100/n S(T), 0)
• The contract includes a short position of n put
  options on the underlying asset with strike
  100/n.

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Interest rate derivatives

• Interest rate options are used to set a limit
  above (cap) or below (floor) to the value of a
  floating coupons.
• A cap/floor is a portfolio of call/put options on
  interest rates, defined on the floating coupon
  schedule
• Each option is called caplet/floorlet
• Libor max(Libor Strike, 0)
• Libor max(Strike Libor, 0)

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Call Put v(t,?)(F Strike)

• Reminding the put-call parity applied to
  cap/floor we have
• Caplet(strike) Floorlet(strike)
• v(t,?)expected coupon strike
• v(t,?)f(t,?,T) strike
• This suggests that the underlying of caplet and
  floorlet are forward rates, instead of spot
  rates.

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Cap/Floor Black formula

• Using Black formula, we have
• Caplet (v(t,tj) v(t,tj1))N(d1) v(t,tj1)
  KN(d2)
• Floorlet
• (v(t,tj1) v(t,tj))N( d1) v(t,tj1) KN(
  d2)
• The formula immediately suggests a replicating
  strategy or a hedging strategy, based on long
  (short) positions on maturity tj and short (long)
  on maturity tji for caplets (floorlets)

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Swaption

• Swaptions are options to enter a payer or
  receiver swap, for a swap rate at a given strike,
  at a future date.
• A payer-swaption provides the right, but not the
  obligation, to enter a payer swap, and
  corresponds to call option, while the receiver
  swaption gives the right, but not the obligation,
  to enter a receiver swap, and corresponds to a
  put option.

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Swaption

• A swaption gives the right, but not the
  obligation, to enter a swap contract at a future
  date tn for swap rate Rs.
• Reset dates tn , tn1,tN for the swap, with
  payments due at dates tn 1 , tn 2,tN 1
• Define ?i ti 1 ti the daycount factors

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The pay-off of a swaption

• A swaption with strike Rs has payoff
• ?i maxR(tnn,N) - Rs ,0
• where R(tnn,N) is the swap rate that will be
  observed at time tn with present value,
• A(tnn,N) maxR(tnn,N) - Rs ,0

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and valuation

• The value of a swaption is computed using
• Swaption A(tn,N) EAmaxR(tnn,N) - Rs ,0
• Assuming the swap rate to be log-normally
  distributed (Swap Market Model), we have Black
  formula
• Swaption A(tn,N) BlackS(tn,N),K,tn,?(n,N)
• or explicitly
• Swaption (v(t,tj) v(t,tN))N(d1)
  ?iv(t,ti) KN(d2)

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